Abstract
In this paper, we define the holomorphic Laplacian for a Frobenius manifold. We give the description of the Laplacian in terms of the prepotential. This will be used to characterize the flat coordinates for the universal unfolding of the function with a simple elliptic singularity. This characterization enables us to solve the so-called “Jacobi’s inversion problem,” i.e. the description of the flat coordinates as the automorphic functions on the period domain w.r.t. the period mapping for the primitive forms. More explicitly the Laplacian relates the flat coordinates with the theta functions on the period domain (see 10.3). About the description of flat coordinates as the automorphic functions, we shall write in a forthcoming paper. The author would like to thank Prof. Claus Hertling and Prof. Atsushi Takahashi for valuable discussions. The auther also thanks the referee for the valuable advices.
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References
B. Dubrovin, Geometry of 2D topological field theories, Integrable Systems and Quantum Groups (ed. by R. Donagi, et al.), Lecture Notes in Math. 1620, Springer-Verlag, 1996 pp. 120–348.
H. Grauert, R. Remmert, Coherent Analytic Sheaves, Grundlehren der mathematischen Wissenschaften 265, Springer-Verlag, 1984.
C. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Math. 151, Cambridge Univ. Press, 2002.
V. G. Kac, Infinite dimensional Lie algebra, Cambridge University Press, third edition, 1995.
A. Klemm, S. Theisen, M.G. Schmidt, Correlation functions for topological Landau-Ginzburg models with c ≤ 3, International Journal of Modern Physics A. 7, No.25 (1992) 6215–6244.
E. Looijenga, Isolated Singular Points on Complete Intersections, London Math. Soc. Lecture Note Series 77, Cambridge University Press, 1984.
Z. Maassarani, On the solution of topological Landau-Ginzburg models with c = 3, Physics Letters B 273 (1991) 457–462.
M. Noumi, Y. Yamada, Notes on the flat structures associated with simple and simply elliptic singularities, Integrable Systems and Algebraic Geometry, Proceedings of the Taniguchi Symposium 1997, World Scientific, 1998.
S. Rosenberg, The Laplacian on a Riemannian Manifold, London Mathematical Society Student Texts 31, 1997.
K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo, Sect. IA Math, 27 (1981) 265–291.
K. Saito, Period Mapping Associated to a Primitive Form, Publ. RIMS. Kyoto Univ. 19 (1983) 1231–1264.
K. Saito, Extended Affine Root System II, Publ. RIMS. Kyoto Univ. 26 (1990) 15–78.
M. Saito, On the structure of Brieskorn lattice, Ann. Inst. Fourier Grenoble 39 (1989) 27–72.
I. Satake, Flat Structure for the Simple Elliptic Singularity of Thpe EĒ6 and Jacobi Form, Proceedings of the Japan Academy, vol.69, Ser. A, No. 7 (1993) 247–251.
I. Satake, Flat Structure and the Prepotential for the Elliptic Root System of Type D (1,1)4 , Topological field theory, primitive forms and related topics (ed. by M. Kashiwara, et al.), Progress in Math. 160, Birkhäuser, 1998, 427–452.
E. Verlinde, N.P. Warner, Topological Landau-Ginzburg matter at c = 3, Physics Letters B 269 (1991) 96–102.
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© 2004 Friedr. Vieweg & Sohn Verlag/GWV Fachverlage GmbH, Wiesbaden
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Satake, I. (2004). The Laplacian for a Frobenius manifold. In: Hertling, K., Marcolli, M. (eds) Frobenius Manifolds. Aspects of Mathematics, vol 36. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-80236-1_12
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DOI: https://doi.org/10.1007/978-3-322-80236-1_12
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